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+/* clogl.c
+ *
+ * Complex natural logarithm
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void clogl();
+ * cmplxl z, w;
+ *
+ * clogl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns complex logarithm to the base e (2.718...) of
+ * the complex argument x.
+ *
+ * If z = x + iy, r = sqrt( x**2 + y**2 ),
+ * then
+ * w = log(r) + i arctan(y/x).
+ *
+ * The arctangent ranges from -PI to +PI.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 7000 8.5e-17 1.9e-17
+ * IEEE -10,+10 30000 5.0e-15 1.1e-16
+ *
+ * Larger relative error can be observed for z near 1 +i0.
+ * In IEEE arithmetic the peak absolute error is 5.2e-16, rms
+ * absolute error 1.0e-16.
+ */
+
+#include <math.h>
+#ifdef ANSIPROT
+static void cchshl ( long double x, long double *c, long double *s );
+static long double redupil ( long double x );
+static long double ctansl ( cmplxl *z );
+long double cabsl ( cmplxl *x );
+void csqrtl ( cmplxl *x, cmplxl *y );
+void caddl ( cmplxl *x, cmplxl *y, cmplxl *z );
+extern long double fabsl ( long double );
+extern long double sqrtl ( long double );
+extern long double logl ( long double );
+extern long double expl ( long double );
+extern long double atan2l ( long double, long double );
+extern long double coshl ( long double );
+extern long double sinhl ( long double );
+extern long double asinl ( long double );
+extern long double sinl ( long double );
+extern long double cosl ( long double );
+void clogl ( cmplxl *, cmplxl *);
+void casinl ( cmplxl *, cmplxl *);
+#else
+static void cchshl();
+static long double redupil();
+static long double ctansl();
+long double cabsl(), fabsl(), sqrtl();
+lnog double logl(), expl(), atan2l(), coshl(), sinhl();
+long double asinl(), sinl(), cosl();
+void caddl(), csqrtl(), clogl(), casinl();
+#endif
+
+extern long double MAXNUML, MACHEPL, PIL, PIO2L;
+
+void clogl( z, w )
+register cmplxl *z, *w;
+{
+long double p, rr;
+
+/*rr = sqrt( z->r * z->r + z->i * z->i );*/
+rr = cabsl(z);
+p = logl(rr);
+#if ANSIC
+rr = atan2l( z->i, z->r );
+#else
+rr = atan2l( z->r, z->i );
+if( rr > PIL )
+ rr -= PIL + PIL;
+#endif
+w->i = rr;
+w->r = p;
+}
+ /* cexpl()
+ *
+ * Complex exponential function
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cexpl();
+ * cmplxl z, w;
+ *
+ * cexpl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns the exponential of the complex argument z
+ * into the complex result w.
+ *
+ * If
+ * z = x + iy,
+ * r = exp(x),
+ *
+ * then
+ *
+ * w = r cos y + i r sin y.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8700 3.7e-17 1.1e-17
+ * IEEE -10,+10 30000 3.0e-16 8.7e-17
+ *
+ */
+
+void cexpl( z, w )
+register cmplxl *z, *w;
+{
+long double r;
+
+r = expl( z->r );
+w->r = r * cosl( z->i );
+w->i = r * sinl( z->i );
+}
+ /* csinl()
+ *
+ * Complex circular sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void csinl();
+ * cmplxl z, w;
+ *
+ * csinl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * w = sin x cosh y + i cos x sinh y.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8400 5.3e-17 1.3e-17
+ * IEEE -10,+10 30000 3.8e-16 1.0e-16
+ * Also tested by csin(casin(z)) = z.
+ *
+ */
+
+void csinl( z, w )
+register cmplxl *z, *w;
+{
+long double ch, sh;
+
+cchshl( z->i, &ch, &sh );
+w->r = sinl( z->r ) * ch;
+w->i = cosl( z->r ) * sh;
+}
+
+
+
+/* calculate cosh and sinh */
+
+static void cchshl( x, c, s )
+long double x, *c, *s;
+{
+long double e, ei;
+
+if( fabsl(x) <= 0.5L )
+ {
+ *c = coshl(x);
+ *s = sinhl(x);
+ }
+else
+ {
+ e = expl(x);
+ ei = 0.5L/e;
+ e = 0.5L * e;
+ *s = e - ei;
+ *c = e + ei;
+ }
+}
+
+ /* ccosl()
+ *
+ * Complex circular cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccosl();
+ * cmplxl z, w;
+ *
+ * ccosl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * w = cos x cosh y - i sin x sinh y.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8400 4.5e-17 1.3e-17
+ * IEEE -10,+10 30000 3.8e-16 1.0e-16
+ */
+
+void ccosl( z, w )
+register cmplxl *z, *w;
+{
+long double ch, sh;
+
+cchshl( z->i, &ch, &sh );
+w->r = cosl( z->r ) * ch;
+w->i = -sinl( z->r ) * sh;
+}
+ /* ctanl()
+ *
+ * Complex circular tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ctanl();
+ * cmplxl z, w;
+ *
+ * ctanl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * sin 2x + i sinh 2y
+ * w = --------------------.
+ * cos 2x + cosh 2y
+ *
+ * On the real axis the denominator is zero at odd multiples
+ * of PI/2. The denominator is evaluated by its Taylor
+ * series near these points.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5200 7.1e-17 1.6e-17
+ * IEEE -10,+10 30000 7.2e-16 1.2e-16
+ * Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
+ */
+
+void ctanl( z, w )
+register cmplxl *z, *w;
+{
+long double d;
+
+d = cosl( 2.0L * z->r ) + coshl( 2.0L * z->i );
+
+if( fabsl(d) < 0.25L )
+ d = ctansl(z);
+
+if( d == 0.0L )
+ {
+ mtherr( "ctan", OVERFLOW );
+ w->r = MAXNUML;
+ w->i = MAXNUML;
+ return;
+ }
+
+w->r = sinl( 2.0L * z->r ) / d;
+w->i = sinhl( 2.0L * z->i ) / d;
+}
+ /* ccotl()
+ *
+ * Complex circular cotangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccotl();
+ * cmplxl z, w;
+ *
+ * ccotl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * sin 2x - i sinh 2y
+ * w = --------------------.
+ * cosh 2y - cos 2x
+ *
+ * On the real axis, the denominator has zeros at even
+ * multiples of PI/2. Near these points it is evaluated
+ * by a Taylor series.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 3000 6.5e-17 1.6e-17
+ * IEEE -10,+10 30000 9.2e-16 1.2e-16
+ * Also tested by ctan * ccot = 1 + i0.
+ */
+
+void ccotl( z, w )
+register cmplxl *z, *w;
+{
+long double d;
+
+d = coshl(2.0L * z->i) - cosl(2.0L * z->r);
+
+if( fabsl(d) < 0.25L )
+ d = ctansl(z);
+
+if( d == 0.0L )
+ {
+ mtherr( "ccot", OVERFLOW );
+ w->r = MAXNUML;
+ w->i = MAXNUML;
+ return;
+ }
+
+w->r = sinl( 2.0L * z->r ) / d;
+w->i = -sinhl( 2.0L * z->i ) / d;
+}
+
+/* Program to subtract nearest integer multiple of PI */
+/* extended precision value of PI: */
+#ifdef UNK
+static double DP1 = 3.14159265160560607910E0;
+static double DP2 = 1.98418714791870343106E-9;
+static double DP3 = 1.14423774522196636802E-17;
+#endif
+
+#ifdef DEC
+static unsigned short P1[] = {0040511,0007732,0120000,0000000,};
+static unsigned short P2[] = {0031010,0055060,0100000,0000000,};
+static unsigned short P3[] = {0022123,0011431,0105056,0001560,};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+#ifdef IBMPC
+static unsigned short P1[] = {0x0000,0x5400,0x21fb,0x4009};
+static unsigned short P2[] = {0x0000,0x1000,0x0b46,0x3e21};
+static unsigned short P3[] = {0xc06e,0x3145,0x6263,0x3c6a};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+#ifdef MIEEE
+static unsigned short P1[] = {
+0x4009,0x21fb,0x5400,0x0000
+};
+static unsigned short P2[] = {
+0x3e21,0x0b46,0x1000,0x0000
+};
+static unsigned short P3[] = {
+0x3c6a,0x6263,0x3145,0xc06e
+};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+static long double redupil(x)
+long double x;
+{
+long double t;
+long i;
+
+t = x/PIL;
+if( t >= 0.0L )
+ t += 0.5L;
+else
+ t -= 0.5L;
+
+i = t; /* the multiple */
+t = i;
+t = ((x - t * DP1) - t * DP2) - t * DP3;
+return(t);
+}
+
+/* Taylor series expansion for cosh(2y) - cos(2x) */
+
+static long double ctansl(z)
+cmplxl *z;
+{
+long double f, x, x2, y, y2, rn, t;
+long double d;
+
+x = fabsl( 2.0L * z->r );
+y = fabsl( 2.0L * z->i );
+
+x = redupil(x);
+
+x = x * x;
+y = y * y;
+x2 = 1.0L;
+y2 = 1.0L;
+f = 1.0L;
+rn = 0.0;
+d = 0.0;
+do
+ {
+ rn += 1.0L;
+ f *= rn;
+ rn += 1.0L;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 + x2;
+ t /= f;
+ d += t;
+
+ rn += 1.0L;
+ f *= rn;
+ rn += 1.0L;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 - x2;
+ t /= f;
+ d += t;
+ }
+while( fabsl(t/d) > MACHEPL );
+return(d);
+}
+ /* casinl()
+ *
+ * Complex circular arc sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void casinl();
+ * cmplxl z, w;
+ *
+ * casinl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Inverse complex sine:
+ *
+ * 2
+ * w = -i clog( iz + csqrt( 1 - z ) ).
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 10100 2.1e-15 3.4e-16
+ * IEEE -10,+10 30000 2.2e-14 2.7e-15
+ * Larger relative error can be observed for z near zero.
+ * Also tested by csin(casin(z)) = z.
+ */
+
+void casinl( z, w )
+cmplxl *z, *w;
+{
+static cmplxl ca, ct, zz, z2;
+long double x, y;
+
+x = z->r;
+y = z->i;
+
+if( y == 0.0L )
+ {
+ if( fabsl(x) > 1.0L )
+ {
+ w->r = PIO2L;
+ w->i = 0.0L;
+ mtherr( "casinl", DOMAIN );
+ }
+ else
+ {
+ w->r = asinl(x);
+ w->i = 0.0L;
+ }
+ return;
+ }
+
+/* Power series expansion */
+/*
+b = cabsl(z);
+if( b < 0.125L )
+{
+z2.r = (x - y) * (x + y);
+z2.i = 2.0L * x * y;
+
+cn = 1.0L;
+n = 1.0L;
+ca.r = x;
+ca.i = y;
+sum.r = x;
+sum.i = y;
+do
+ {
+ ct.r = z2.r * ca.r - z2.i * ca.i;
+ ct.i = z2.r * ca.i + z2.i * ca.r;
+ ca.r = ct.r;
+ ca.i = ct.i;
+
+ cn *= n;
+ n += 1.0L;
+ cn /= n;
+ n += 1.0L;
+ b = cn/n;
+
+ ct.r *= b;
+ ct.i *= b;
+ sum.r += ct.r;
+ sum.i += ct.i;
+ b = fabsl(ct.r) + fabs(ct.i);
+ }
+while( b > MACHEPL );
+w->r = sum.r;
+w->i = sum.i;
+return;
+}
+*/
+
+
+ca.r = x;
+ca.i = y;
+
+ct.r = -ca.i; /* iz */
+ct.i = ca.r;
+
+ /* sqrt( 1 - z*z) */
+/* cmul( &ca, &ca, &zz ) */
+zz.r = (ca.r - ca.i) * (ca.r + ca.i); /*x * x - y * y */
+zz.i = 2.0L * ca.r * ca.i;
+
+zz.r = 1.0L - zz.r;
+zz.i = -zz.i;
+csqrtl( &zz, &z2 );
+
+caddl( &z2, &ct, &zz );
+clogl( &zz, &zz );
+w->r = zz.i; /* mult by 1/i = -i */
+w->i = -zz.r;
+return;
+}
+ /* cacosl()
+ *
+ * Complex circular arc cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cacosl();
+ * cmplxl z, w;
+ *
+ * cacosl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ *
+ * w = arccos z = PI/2 - arcsin z.
+ *
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5200 1.6e-15 2.8e-16
+ * IEEE -10,+10 30000 1.8e-14 2.2e-15
+ */
+
+void cacosl( z, w )
+cmplxl *z, *w;
+{
+
+casinl( z, w );
+w->r = PIO2L - w->r;
+w->i = -w->i;
+}
+ /* catanl()
+ *
+ * Complex circular arc tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void catanl();
+ * cmplxl z, w;
+ *
+ * catanl( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ * 1 ( 2x )
+ * Re w = - arctan(-----------) + k PI
+ * 2 ( 2 2)
+ * (1 - x - y )
+ *
+ * ( 2 2)
+ * 1 (x + (y+1) )
+ * Im w = - log(------------)
+ * 4 ( 2 2)
+ * (x + (y-1) )
+ *
+ * Where k is an arbitrary integer.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5900 1.3e-16 7.8e-18
+ * IEEE -10,+10 30000 2.3e-15 8.5e-17
+ * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
+ * had peak relative error 1.5e-16, rms relative error
+ * 2.9e-17. See also clog().
+ */
+
+void catanl( z, w )
+cmplxl *z, *w;
+{
+long double a, t, x, x2, y;
+
+x = z->r;
+y = z->i;
+
+if( (x == 0.0L) && (y > 1.0L) )
+ goto ovrf;
+
+x2 = x * x;
+a = 1.0L - x2 - (y * y);
+if( a == 0.0L )
+ goto ovrf;
+
+#if ANSIC
+t = atan2l( 2.0L * x, a ) * 0.5L;
+#else
+t = atan2l( a, 2.0 * x ) * 0.5L;
+#endif
+w->r = redupil( t );
+
+t = y - 1.0L;
+a = x2 + (t * t);
+if( a == 0.0L )
+ goto ovrf;
+
+t = y + 1.0L;
+a = (x2 + (t * t))/a;
+w->i = logl(a)/4.0;
+return;
+
+ovrf:
+mtherr( "catanl", OVERFLOW );
+w->r = MAXNUML;
+w->i = MAXNUML;
+}
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